We show that jigsaw puzzles, edge-matching puzzles, and
polyomino packing puzzles are all NP-complete. Furthermore,
we show direct equivalences between these three types of puzzles: any puzzle
of one type can be converted into an equivalent puzzle of any other type.

The last paragraph of the introduction asks about polyomino packing puzzles where each piece has just logarithmic area. Michael Brand (2007) has proved such puzzles NP-complete by following a similar approach to this paper, but going directly from 3-partition to polyomino packing.